Quartal and quintal harmony
In music, quartal harmony is the building of harmonic structures with a distinct preference for the intervals of the perfect fourth, the augmented fourth and the diminished fourth. Quintal harmony is harmonic structure preferring the perfect fifth, the augmented fifth and the diminished fifth.
Use of the terms quartal and quintal arises from a contrast, compositional or perceptual, with traditional tertian harmonic constructions. Listeners familiar with music of the (European) common practice period perceive tonal music as that which uses major and minor chords and scales, wherein both the major third and minor third constitute the basic structural elements of the harmony.
Quintal harmony (the harmonic layering of fifths specifically) is a lesser-used term, and since the fifth is the inversion or complement of the fourth, it is usually considered indistinct from quartal harmony. Indeed, a circle of fifths can be arranged in fourths (G→C→F→B♭ etc. are fifths when played downwards and fourths when played upwards); this is the reason that modern theoreticians may speak of a "circle of fourths".
- 1 Analysis
- 2 History
- 3 Examples of quartal pieces
- 4 See also
- 5 References
- 6 Further reading
- 7 External links
The concept of quartal harmony outlines a formal harmonic structure based on the use of the interval of a perfect fourth to form chords. The fourth, thus, substitutes for the third as used in chords based on major and minor thirds. Although the fourth replaces the third in chords, quartal harmony rarely replaces tertian harmony in full works. Instead, the two types of harmony are found side-by-side. Since the distance between the lower and the higher notes of a stack of two perfect fourths is a minor seventh and this interval inverts to a major second, quartal harmony necessarily also includes these intervals.
A quartal chord composed of the notes C – F – B♭ may be regarded using traditional theory as a C dominant seventh chord (with an omitted fifth) in the midst of a 4–3 suspension, or as C7sus4 (see suspended chord), where the fourth does not require resolution. Fsus4, a suspended second-inversion chord, would also be a plausible label. Extending quartal chords to four or more notes generates still more possibilities of a similar nature. The four-note chord C – F – B♭ – E♭ can be interpreted as a C minor chord with a minor seventh and embellishing fourth (Cm7add4 or Cm11), or as an inversion of an E-flat major chord with a second-suspension and embellishing sixth—E♭sus2(add6), among other interpretations.
The question of which strategy of analysis is advisable is hard to answer since it is refined by the particular details: given one interpretation, and the progression of harmony through the preceding and following chords, and the overall musical development, is there a comprehensible and audibly functional meaning to the interpretation? It is important to question whether these suspensions, chromatic chords and altered chords are resolved as part of the functional harmony or whether they remain non-functional and unresolved.
In the Middle Ages, simultaneous notes a fourth apart were heard as a consonance. During the common practice period (between about 1600 and 1900), this interval came to be heard either as a dissonance (when appearing as a suspension requiring resolution in the voice leading) or as a consonance (when the tonic of the chord appears in parts higher than the fifth of the chord). In the later 19th century, during the breakdown of tonality in classical music, all intervallic relationships were once again reassessed. Quartal harmony was developed in the early 20th century as a result of this breakdown and reevaluation of tonality.
The Tristan chord is made up of the notes F♮, B♮, D♯ and G♯ and is the very first chord heard in Wagner's opera Tristan und Isolde. The bottom two notes make up an augmented fourth; the upper two make up a perfect fourth. This layering of fourths in this context has been seen as highly significant. The chord had been found in earlier works (Vogel 1962, 12; Nattiez 1990,[page needed]) (notably Beethoven's Piano Sonata No. 18) but Wagner's use was significant, first because it is seen as moving away from traditional tonal harmony and even towards atonality, and second because with this chord Wagner actually provoked the sound or structure of musical harmony to become more predominant than its function, a notion which was soon after to be explored by Debussy and others (Erickson 1975,[page needed]). Beethoven's use of the chord is of short duration and it resolves in the accepted manner; whereas Wagner's use lasts much longer and resolves in a highly unorthodox manner for the time. Despite the layering of fourths, it is rare to find musicologists identifying this chord as "quartal harmony" or even as "proto-quartal harmony", since Wagner's musical language is still essentially built on thirds, and even an ordinary dominant seventh chord can be laid out as augmented fourth plus perfect fourth (F-B-D-G). Wagner's unusual chord is really a device to draw the listener into the musical-dramatic argument with which the composer in presenting us. However, fourths become important later in the opera, especially in the melodic development.
At the beginning of the 20th century, fourth-based chords finally became an important element of harmony.
Scriabin used a self-developed system of transposition using fourth-chords, like his Mystic chord in his Piano Sonata No. 6. Scriabin wrote this chord in his sketches alongside other quartal passages and more traditional tertian passages, often passing between systems, for example widening the six-note quartal sonority (C – F♯ – B♭ – E – A – D) into a seven-note chord (C – F♯ – B♭ – E – A – D – G).
In the 1897 work Paul Dukas's The Sorcerer's Apprentice, we hear a rising repetition in fourths, as the tireless work of out-of-control walking brooms causes the water level in the house to "rise and rise". Quartal harmony in Ravel's Sonatine and Ma mère l'oye would follow a few years later.
20th- and 21st-century classical music
Composers who use the techniques of quartal harmony include Claude Debussy, Francis Poulenc, Alexander Scriabin, Alban Berg, Leonard Bernstein, Arnold Schoenberg, Igor Stravinsky, and Anton Webern (Herder 1987, 78).
Arnold Schoenberg's Chamber Symphony Op. 9 (1906) displays quartal harmony. The work begins not from tonal harmony, but instead begins with a fictitious[clarification needed] tonal centre: the first measures construct a five-part fourth chord with the notes C – F – B♭ – E♭ – A♭ distributed over several instruments. The composer then picks out this vertical quartal harmony in a horizontal sequence of fourths from the horns, eventually leading to a passage of triadic quartal harmony (i.e., chords of three notes, each layer a fourth apart).
Schoenberg was also one of the first to write on the theoretical consequences of this harmonic innovation. In his Theory of Harmony (Harmonielehre) of 1911 he wrote: "The construction of chords by superimposing fourths can lead to a chord that contains all the twelve notes of the chromatic scale; hence, such construction does manifest a possibility for dealing systematically with those harmonic phenomena that already exist in the works of some of us: seven, eight, nine, ten, eleven, and twelve-part chords... But the quartal construction makes possible, as I said, accommodation of all phenomena of harmony" (Schoenberg 1978, 406–407). Other examples of quartal harmony appear in Schoenberg's String Quartet No. 1.
Webern, Ives, and Bartók
For Anton Webern, the importance of quartal harmony lay in the possibility of building new sounds. In 1912, he wrote, "With alteration the fourth-chord never need belong to tonal harmony, but can be free of all tonal relationships."[this quote needs a citation] After hearing Schoenberg's Chamber Symphony, Webern wrote "You must write something like that, too!" (The Path to the New Music, p.48. "So was mußt du auch machen!"[full citation needed]) Shortly after, he wrote his Four Pieces for Violin and Piano Op. 7, using quartal harmony as a formal principle, which was also used in later works.
Uninfluenced by the theoretical and practical work of the Second Viennese School, the American Charles Ives meanwhile wrote in 1906 a song called "The Cage" (No. 64 of his collection, 114 songs), in which the piano part contained four-part fourth chords accompanying a vocal line which moves in whole tones.
Other 20th-century composers, like Béla Bartók with his piano work Mikrokosmos and Music for Strings, Percussion and Celesta, as well as Paul Hindemith, Carl Orff and Igor Stravinsky, employed quartal harmony in their work. These composers joined Romantic elements with Baroque music, folk songs and their peculiar rhythm and harmony with the open harmony of fourths and fifths.
Hindemith constructed large parts of his symphonic work Symphony: Mathis der Maler by means of fourth and fifth intervals. These steps are a restructuring of fourth chords (C – D – G becomes the fourth chord D – G – C), or other mixtures of fourths and fifths (D♯ – A♯ – D♯ – G♯ – C♯ in measure 3 of the example). Hindemith was, however, not a proponent of an explicit quartal harmony. In his 1937 writing Unterweisung im Tonsatz (The Craft of Musical Composition, Hindemith 1937), he wrote that "notes have a family of relationships, that are the bindings of tonality, in which the ranking of intervals is unambiguous," so much so, indeed, that in the art of triadic composition "...the musician is bound by this, as the painter to his primary colours, the architect to the three dimensions." He lined up the harmonic and melodic aspects of music in a row in which the octave ranks first, then the fifth and the third, and then the fourth. "The strongest and most unique harmonic interval after the octave is the fifth, the prettiest nevertheless is the third by right of the chordal effects of its Combination tones."
In his Theory of Harmony (Schoenberg 1978, 407): "Besides myself my students Dr. Anton Webern and Alban Berg have written these harmonies (fourth chords), but also the Hungarian Béla Bartók or the Viennese Franz Schreker, who both go a similar way to Debussy, Dukas and perhaps also Puccini, are not far off.
British composer Michael Tippett also employed quartal harmonies extensively in works from his middle period. Examples are his Piano Concerto and the opera The Midsummer Marriage. An almost constant quartal harmony is used by Bertold Hummel in his Second Symphony of 1966. A similarly obvious example is the work of Mieczysław Weinberg. Hermann Schroeder alternated in his works using fragments of Gregorian Chant between quintal and quartal harmony. Also the Polish composer Witold Lutosławski devised a use that allows many harmonic combinations to be applied to a single part, having several combinations that may be tried against it, like fourths with whole tones, tritones with semitones, or other possibilities.
In the first movement of Olivier Messiaen's Turangalîla-Symphonie, a six-note combination is constructed in pieces from fourths and tritones, much like in the music of Schoenberg and Scriabin. Much of Messiaen's work applies quartal harmony, moderated by his development of what he called "Modes of limited transposition".
A preference for quartal harmony is present in the works of Leo Brouwer (10 Etudes for Guitar), Robert Delanoff (Zwiegespräche für Orgel), Ivan Vïshnegradsky, Tōru Takemitsu (Cross Hatch) and Hanns Eisler (Hollywood-Elegy). In the 1960s, the use of tone clusters juxtaposing minor and major seconds pushed aside quartal harmony somewhat. The orchestral work of György Ligeti, Atmosphères of 1961, makes extensive use of such sounds. The works of the Filipino composer Elisio Pajaro (1915–1984) are characterised by quartal and quintal harmonies, as well as by dissonant counterpoint and polychords (Kasilag 2001).
As a transition to the history of jazz, George Gershwin may be mentioned. In the first movement of his Concerto in F altered fourth chords descend chromatically in the right hand with a chromatic scale leading upward in the left hand.
The style of jazz, having an eclectic harmonic orbit, was in its early days overtaken (until perhaps the Swing of the 1930s) by the vocabulary of 19th century European music. Important influences come thereby from opera, operetta, military bands as well as from the piano music of Classical and Romantic composers, and even that of the Impressionists. Jazz musicians had a clear interest in harmonic richness of colour, for which quartal harmony provided possibilities, as used by pianists and arrangers like Jelly Roll Morton, Duke Ellington, Art Tatum, Bill Evans (Hester 2000, 199) Milt Buckner (Hester 2000, 199) Chick Corea (Herder 1987, 78; Scivales 2005, 203) Herbie Hancock (Herder 1987, 78; Scivales 2005, 203) and especially McCoy Tyner (Herder 1987, 78; Scivales 2005, 205). Nevertheless, the older jazz usually handled fourths in the customary manner (as a suspension needing resolution).
Bebop brought an aesthetic change to modern jazz: the chords which before had a relative identity (as major and minor, dominant, etc.) gave way to block transpositions, with a fleeting, smooth flowing tonality, having the colours of chords blurred and strongly ambiguous. A prevalent example for this is the beloved ii-V-I cadence of modern jazz.
In the figure to the right, a traditional cadence is contrasted with a cadence where a substitution has been made in one of the inner voices. The inner voice still exhibits normal voice leading but within the extended harmony of jazz. The multiplicity of possibilities available can be used as a framework for improvisation. In addition, compositions of this time often had a frantic tempo, allowing more leeway in the harmony of fleeting chords (because they are not sounding for very long). Quartal harmony was employed throughout the jazz of the 1940s.
The hard bop of the 1950s made new applications of quartal harmony accessible to jazz. Quintet writing in which two brass instruments (commonly trumpet and saxophone) may proceed in fourths, while the piano (as a uniquely harmonic instrument) lays down chords, but sparsely, only hinting at the intended harmony. This style of writing, in contrast with that of the previous decade, preferred a moderate tempo. Thin-sounding unison bebop horn sections occur frequently, but these are balanced by bouts of very refined polyphony such as is found in cool jazz.
On his watershed record Kind of Blue, Miles Davis with pianist Bill Evans used a chord consisting of three perfect fourth intervals and a major third on the composition "So What". This particular voicing is sometimes referred to as a So What chord, and can be analyzed (without regard for added sixths, ninths, etc.) as a minor seventh with the root on the bottom, or as a major seventh with the third on the bottom (Levine 1989, 97).
From the outset of the 1960s, the employment of quartal possibilities had become so familiar that the musician now felt the fourth chord existed as a separate entity, self standing and free of any need to resolve. The pioneering of quartal writing in later jazz and rock, like the pianist McCoy Tyner's work with saxophonist John Coltrane's "classic quartet", was influential throughout this epoch. Oliver Nelson was also known for his use of fourth chord voicings (Corozine 2002, 12). Floyd claims that the "foundation of 'modern quartal harmony'" began in the era when the Charlie Parker–influenced John Coltrane added classically trained pianists Bill Evans and McCoy Tyner to his ensemble (Floyd 2004, 4).
Jazz guitarists cited as using chord voicings using quartal harmony include Johnny Smith, Tal Farlow, Chuck Wayne, Barney Kessel, Joe Pass, Jimmy Raney, Wes Montgomery, however all in a traditional manner, as major 9th, 13th and minor 11th chords (Floyd 2004, 4) (an octave and fourth equals an 11th). Jazz guitarists cited as using modern quartal harmony include Jim Hall (especially Sonny Rollins's The Bridge), George Benson ("Skydive"), Pat Martino, Jack Wilkins ("Windows"), Joe Diorio, Howard Roberts ("Impressions"), Kenny Burrell ("So What"), Wes Montgomery ("Little Sunflower"), Henry Johnson, Russell Malone, Jimmy Bruno, Howard Alden, Paul Bollenback, Mark Whitfield, and Rodney Jones (Floyd 2004, 4).
Quartal harmony was also explored as a possibility under new experimental scale models as they were "discovered" by jazz. Musicians began to work extensively with the so-called church modes of old European music, and they became firmly situated in their compositional process. Jazz was well-suited to incorporate the medieval use of fourths to thicken lines into its improvisation. The pianists Herbie Hancock, and Chick Corea are two musicians well known for their modal experimentation. Around this time, a style known as free jazz also came into being, in which quartal harmony had extensive use due to the wandering nature of its harmony.
Between these intensive experiments with quartal harmony, the search for new applications for it in jazz was quickly exhausted. Around 1970, quartal harmony had become part of the canon of everyday practice. In jazz, the way chords were built from a scale came to be called voicing, and specifically quartal harmony was referred to as fourth voicing.
Thus when the m11 and the dominant 7th sus (9sus above) chords in quartal voicings are used together they tend to "blend into one overall sound" sometimes referred to as modal voicings, and both may be applied where the m11 chord is called for during extended periods such as the entire chorus (Boyd 1997, 95).
Quartal and quintal harmony have been used by Robert Fripp, the rhythm guitarist of King Crimson. Fripp dislikes minor thirds and especially major thirds in equal temperament tuning, which is used by non-experimental guitars. Of course, just intonation's perfect octaves, perfect fifths, and perfect fourths are well approximated in equal temperament tuning, and perfect fifths and octaves are highly consonant intervals. Fripp builds chords using perfect fifths, fourths, and octaves in his new standard tuning (NST), a regular tuning having perfect fifths between its successive open-strings. (Mulhern 1986, )[page needed]
Examples of quartal pieces
- Sonata for Alto Saxophone and Piano (Lewis 1985, 443)
- Sonata for Piano, op. 1 (Lambert 1996, 118)
- Wozzeck (Lambert 1996, 118; Reisberg 1975, 344–46)
- Of Mice and Men (Bick 2005, 446, 448, 451)
- "La cathédrale engloutie", beginning and ending (Reisberg 1975, 343–44)
- Suite for Piano
- Piano Sonata No. 7 (Kroeger 1969)
- 12 American Preludes, Prelude #7
- "Donde habite el olvido" (Kulp 2006, 207)
- Bacchanalia for Band (Spieth 1978)
- "The Cage" (1906) (Carr 1989, 135; Lambert 1990, 44; Lambert 1996, 118; Murphy 2008, 179, 181, 183, 185–86, 190–91; Reisberg 1975, 344–45; Scott 1994, 458)
- Central Park in the Dark (Scott 1994, 458)
- "Harpalus" (Scott 1994, 458)
- Psalm 24, verse 5 (Lambert 1990, 67; Scott 1994, 458)
- Psalm 90 (Scott 1994, 458)
- "Walking" (Scott 1994, 458)
- String Quartet No. 2, Adagio (Cowell 1956, 243)
- Sonatina for flute & piano, Op. 76 (Cardew-Fanning n.d.)
- Clarinet Concerto (Archibald 1969, 825)
- Ricercare for Orchestra (Archibald 1969, 825)
- King Midas, cantata (Sjoerdsma 1972)
- Le fils des étoiles (Carpenter n.d.; Reisberg 1975, 347)
- Diatonic Study (1914) (Stein 1979, 18)
- Suite No. 3 for Piano (Dickinson 1963)
- Piano Sonata (Swayne 2002, 285–87, 290)
- Klavierstück IX (Reisberg 1975, 349–50)
- "Saw a Grave" (Moe 1981–82, 70)
- Variations for Piano, Op. 27 (Reisberg 1975, 348)
- Kind of Blue (Josh n.d.)
- Emerson, Lake & Palmer
- Frank Zappa
- Anon. (n.d. ) "Rook", ChalkHills.org. Analysis and guitar tab.
- Archibald, Bruce (1969). "Variations for Cello and Orchestra (1966) by Walter Piston; Concerto for Clarinet and Orchestra (1967) by Walter Piston; Ricercare for Orchestra (1967) by Walter Piston". Notes, second series 25, no. 4 (June): 824–26.
- Benward, Bruce, and Nadine Saker (2009). Music in Theory and Practice, Vol. II. ISBN 978-0-07-310188-0.[full citation needed]
- Bick, Sally (2005). "Of Mice and Men: Copland, Hollywood, and American Musical Modernism". American Music 23, no. 4 (Winter): 426–72.
- Boyd, Bill (1997). Jazz Chord Progressions.[full citation needed] ISBN 0-7935-7038-7.
- Cardew-Fanning, Neil. "Sonatina for flute & piano, Op. 76", AllMusic.com.
- Carpenter, Alexander (n.d.). "Le fils des étoiles, Chaldean pastoral, 3 preludes for piano", AllMusic.com.
- Carr, Cassandra I. (1989). "Charles Ives’s Humor as Reflected in His Songs". American Music 7, no. 2 (Summer): 123–39.
- Corozine, Vince (2002). Arranging Music for the Real World: Classical and Commercial Aspects. Pacific, MO: Mel Bay. ISBN 0-7866-4961-5. OCLC 50470629.
- Corson, Langdon, and Roy Christensen (1984). Arnold Schoenberg's Woodwind Quintet, op. 26: Background and Analysis. Nashville: Gasparo Co.
- Cowell, Henry (1956). "Current Chronicle: United States: New York" The Musical Quarterly 42, no. 2 (April): 240–44.
- Dickinson, Peter (1963). "Suite for Piano No 3 by Nikos Skalkottas". The Musical Times 104, no. 1443 (May): 357.
- Domek, Richard C. (1979). "Some Aspects of Organization in Schoenberg's Book of the Hanging Gardens, opus 15". College Music Symposium 19, no. 2 (Fall): 111–28.
- Erickson, Robert (1975). Sound Structure in Music. Oakland, California: University of California Press. ISBN 0-520-02376-5.
- Floyd, Tom (2004). Quartal Harmony & Voicings for Guitar.[full citation needed] ISBN 0-7866-6811-3
- Herder, Ronald (1987). 1000 Keyboard Ideas.[full citation needed] ISBN 0-943748-48-8.
- Hindemith, Paul (1937). Unterweisung im Tonsatz.[full citation needed]
- Hester, Karlton E. (2000). From Africa to Afrocentric Innovations Some Call "Jazz": The Creation of Free, Fusion and Reconstructive Modern Styles (1950–2000)[full citation needed]. ISBN 1-58684-054-1.
- Josh (n.d.) "4th Chords and Quartal Harmony", FretterVerse.com.
- Kasilag, Lucrecia R. (2001). "Pajaro, Eliseo (Morales)". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.
- Kostka, Stefan, Dorothy Payne, and Byron Almén (2013). Tonal Harmony with an Introduction to Twentieth-Century Music (seventh ed.). New York: McGraw-Hill. pp. 38–53. ISBN 978-0-07-131828-0.
- Kroeger, Karl (1969). "Caspar Diethelm: Klaviersonate VII". Notes, second series 26, no. 2 (December): 363.
- Kulp, Jonathan (2006). "Carlos Guastavino: A Re-Evaluation of His Harmonic Language". Latin American Music Review / Revista de Música Latinoamericana 27, no. 2 (Autumn-Winter): 196–219.
- Lambert, J. Philip (1990). "Interval Cycles as Compositional Resources in the Music of Charles Ives". Music Theory Spectrum 12, no. 1 (Spring): 43–82.
- Lambert, J. Phillip (1996). "Ives and Berg: 'Normative' Procedures and Post-Tonal Alternatives". In Charles Ives and the Classical Tradition, edited by Geoffrey Block and J. Peter Burkholder, 105–30. New Haven: Yale University Press.
- Levine, Mark (1989). The Jazz Piano Book. Petaluma, CA: Sher Music Co.. ISBN 0-9614701-5-1.
- Lewis, Robert Hall (1985). "New Music Festival 1985: Bowling Green State University". Perspectives of New Music 24, no. 1 (Fall–Winter): 440–43.
- Leyva, Jesse (2010). "Carlos Chávez: An Examination of His Compositional Style with a Conductor’s Analysis of Sinfonía India as Arranged for Concert Band by Frank Erickson". DMA diss. Tempe: Arizona State University.
- Macon, Edward L (1997). Rocking the Classics: English Progressive Rock and the Counterculture. Oxford and New York: Oxford University Press. ISBN 0-19-509887-0..
- Moe, Orin (1981–82). "The Songs of Howard Swanson". Black Music Research Journal 2:57–71.
- Morrison, S. (1998). "Skryabin and the Impossible". Journal of the American Musicological Society 51, no. 2.[page needed]
- Mulhern, Tom (1986). "On the Discipline of Craft and Art: An Interview with Robert Fripp". Guitar Player. 20 (January): 88–103. Retrieved 8 January 2013.
- Murphy, Scott (2008). "A Composite Approach to Ives’s 'Cage'". Twentieth-Century Music 5:179–93.
- Murphy, Howard Ansley, Robert A. Melcher, and Willard F. Warch, eds. (1973). Music for Study: A Source Book of Excerpts, second edition. Englewood Cliffs, NJ: Prentice-Hall. ISBN 0-13-607515-0.
- Nattiez, Jean-Jacques (1990). Music and Discourse: Toward a Semiology of Music, translated by Carolyn Abbate. Princeton, N.J.: Princeton University Press. ISBN 0-691-02714-5.
- Orbón, Julián (1987). "Las sinfonías de Carlos Chávez." (part 2). Pauta: Cuadernos de teoría y crítica musical 6, no. 22 (April–June): 81–91.
- Reisberg, Horace (1975). "The Vertical Dimension in Twentieth-Century Music". In Aspects of 20th Century Music, Gary E. Wittlich, coordinating editor, 322–87. Englewood Cliffs, NJ: Prentice-Hall. ISBN 0-13-049346-5.
- Rubin, Justin Henry (2005). "Quartal Harmony". University of Minnesota Duluth website d.umn.edu (accessed 26 April 2012).
- Sanderson, Blair (n.d.). "Arnold Schoenberg: Verklärte Nacht; Chamber Symphony; Variations; 5 Pieces; 6 Songs; Erwartung", AllMusic.com.
- Schoenberg, Arnold (1922). Harmonielehre, third edition. Vienna: Universal Edition.
- Schoenberg, Arnold (1978). Theory of Harmony, translated by Roy E. Carter, based on the third edition (1922). Berkeley and Los Angeles: University of California Press. ISBN 0-520-04945-4 (cloth); ISBN 0-520-04944-6 (pbk).
- Scivales, Riccardo (2005). Jazz Piano—The Left Hand.[full citation needed] ISBN 1-929009-54-2.
- Scott, Ann Besser (1994). "Medieval and Renaissance Techniques in the Music of Charles Ives: Horatio at the Bridge?" The Musical Quarterly 78, no. 3 (Autumn): 448–78.
- Sjoerdsma, Richard Dale (1972). "King Midas; A Cantata for Voices and Piano on 10 Poems of Howard Moss by Ned Rorem". Notes, second series 28, no. 4 (June): 782.
- Spieth, Donald (1978). "Bacchanalia for Band by Walter S. Hartley". Notes, second series 34, no. 4 (June): 974.
- Stein, Leon (1979). Structure & Style: The Study and Analysis of Musical Forms, second expanded edition. Princeton, N.J.: Summy-Birchard Music. ISBN 978-0-87487-164-7.
- Swayne, Steve (2002). "Sondheim's Piano Sonata". Journal of the Royal Musical Association 127, no. 2:258–304.
- Vogel, Martin (1962). Der Tristan-Akkord und die Krise der modernen Harmonielehre[full citation needed]
- Whitesell, Lloyd (2008). The Music of Joni Mitchell. Oxford and New York: Oxford University Press. ISBN 978-0-19-530757-3.
- Baker, David N. (1983). Jazz Improvisation. Bloomington: Frangipani. ISBN 0-89917-397-7.
- Persichetti, Vincent (1961). Twentieth-century Harmony: Creative Aspects and Practice. New York: W. W. Norton. ISBN 0-393-09539-8. OCLC 398434.
- Rosenthal, David H. (1993). Hard Bop, Jazz and Black Music 1955–1965. New York: Oxford University Press. ISBN 0-19-508556-6.
- Quartalharmony with notes and listening examples
- Quartal voicing for the guitar
- Program notes for Arnold Schoenberg's Chamber Symphony for 15 Solo Instruments op. 9